643 research outputs found
Composite Cyclotomic Fourier Transforms with Reduced Complexities
Discrete Fourier transforms~(DFTs) over finite fields have widespread
applications in digital communication and storage systems. Hence, reducing the
computational complexities of DFTs is of great significance. Recently proposed
cyclotomic fast Fourier transforms (CFFTs) are promising due to their low
multiplicative complexities. Unfortunately, there are two issues with CFFTs:
(1) they rely on efficient short cyclic convolution algorithms, which has not
been investigated thoroughly yet, and (2) they have very high additive
complexities when directly implemented. In this paper, we address both issues.
One of the main contributions of this paper is efficient bilinear 11-point
cyclic convolution algorithms, which allow us to construct CFFTs over
GF. The other main contribution of this paper is that we propose
composite cyclotomic Fourier transforms (CCFTs). In comparison to previously
proposed fast Fourier transforms, our CCFTs achieve lower overall complexities
for moderate to long lengths, and the improvement significantly increases as
the length grows. Our 2047-point and 4095-point CCFTs are also first efficient
DFTs of such lengths to the best of our knowledge. Finally, our CCFTs are also
advantageous for hardware implementations due to their regular and modular
structure.Comment: submitted to IEEE trans on Signal Processin
Mixing for progressions in non-abelian groups
We study the mixing properties of progressions ,
of length three and four in a model class of finite
non-abelian groups, namely the special linear groups over a finite
field , with bounded. For length three progressions , we
establish a strong mixing property (with error term that decays polynomially in
the order of ), which among other things counts the number of such
progressions in any given dense subset of , answering a question
of Gowers for this class of groups. For length four progressions
, we establish a partial result in the case if the
shift is restricted to be diagonalisable over the field, although in this
case we do not recover polynomial bounds in the error term. Our methods include
the use of the Cauchy-Schwarz inequality, the abelian Fourier transform, the
Lang-Weil bound for the number of points in an algebraic variety over a finite
field, some algebraic geometry, and (in the case of length four progressions)
the multidimensional Szemer\'edi theorem.Comment: 31 pages, no figures, to appear, Forum of Mathematics, Sigma. Referee
suggestions incorporate
Number theoretic techniques applied to algorithms and architectures for digital signal processing
Many of the techniques for the computation of a two-dimensional convolution of a small fixed window with a picture are reviewed. It is demonstrated that Winograd's cyclic convolution and Fourier Transform Algorithms, together with Nussbaumer's two-dimensional cyclic convolution algorithms, have a common general form. Many of these algorithms use the theoretical minimum number of general multiplications. A novel implementation of these algorithms is proposed which is based upon one-bit systolic arrays. These systolic arrays are networks of identical cells with each cell sharing a common control and timing function. Each cell is only connected to its nearest neighbours. These are all attractive features for implementation using Very Large Scale Integration (VLSI). The throughput rate is only limited by the time to perform a one-bit full addition. In order to assess the usefulness to these systolic arrays a 'cost function' is developed to compare them with more conventional techniques, such as the Cooley-Tukey radix-2 Fast Fourier Transform (FFT). The cost function shows that these systolic arrays offer a good way of implementing the Discrete Fourier Transform for transforms up to about 30 points in length. The cost function is a general tool and allows comparisons to be made between different implementations of the same algorithm and between dissimilar algorithms. Finally a technique is developed for the derivation of Discrete Cosine Transform (DCT) algorithms from the Winograd Fourier Transform Algorithm. These DCT algorithms may be implemented by modified versions of the systolic arrays proposed earlier, but requiring half the number of cells
Multi-marginal optimal transport: theory and applications
Over the past five years, multi-marginal optimal transport, a generalization
of the well known optimal transport problem of Monge and Kantorovich, has begun
to attract considerable attention, due in part to a wide variety of emerging
applications. Here, we survey this problem, addressing fundamental theoretical
questions including the uniqueness and structure of solutions. The (partial)
answers to these questions uncover a surprising divergence from the classical
two marginal setting, and reflect a delicate dependence on the cost function.
We go one to describe two applications of the multi-marginal problem.Comment: Typos corrected and minor changes to presentatio
Generalised Type-II Hybrid Automatic Repeat Request Schemes and KM Codes
Abstract not provided
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